Fred J. Schaal                 Lane Tech High School
                               Addison & Western
                               Chicago, IL 60613
                               1-312-880-8100 Math Dept.

     Make use of concentric circles and angles (in excess of 180 degrees) to 
construct a sketch of the naked-eye-visible planets of the Solar System.  (What 
this amounts to is polar coordinates, but I do not mention it.) 

     Graph paper, compass, protractor, and table of astronomical data: 
heliocentric longitude, in counter-clockwise degrees, and the semi-major axis 
(radius, for practical purposes) of the orbit, in astronomical units, for the 
planets Venus, Mars, Earth, Jupiter and Saturn. 

     Draw concentric circles on the board--one for each planet's radius (semi-
major axis).  Plot the position of each planet on its respective circle.  Explain 
how the figure indicates if, where and when a given planet will be visible on the 
date in question. 

     The source for the heliocentric coordinates is the current issue of Astronomy 
Magazine--July 86--for all but the planet earth. (How strange!)  To get this 
terrestrial data I phoned the Adler Planetarium. It is necessary to interpolate 
between dates to find the coordinate for Mars.  
      The source for the lengths of the semi-major axes (radii) of the orbits of 
the planets is an old astronomy book from college, but any almanac or general-
information-type book should suffice.  

      Using standard, generic, drug-store, (quarter inch per side of a square) 
graph paper, a scale of two units per astronomical unit is good.  The Sun is 
placed 16 units over and 7 units down from the upper left corner: the longer edge 
is horizontal.  (It just so happens that for the date in question most of the 
planets lie in the bottom half of the concentric circle system.  This means that 
only halves of the big circles--Saturn and Jupiter--have to be drawn--the bottom 
halves. This allows for a larger scale for the whole sketch.  All of the circles 
for Venus, Earth and Mars can and should be drawn.)  A ray is drawn due East, (to 
the right, parallel to the long edge of the graph paper), from the Sun, to serve 
as both a distance scale and an angle reference.  The direction of the ray is zero 
degrees of heliocentric longitude.  (Technically it points towards the Vernal 
Equinox.)  Each planet's position is found by moving counter clockwise about the 
proper circle from this reference ray.  

    Explaining visibility is beyond the scope of a one page write-up of this mini-
teach.  With some thought, it should become obvious to anyone but the most casual 

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